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Arithmetic
Contents
Rational Numbers
Operations
Place Values
Place Values
Digit: A number that names a place value location (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).
Periods: Groups of three digits separated by commas. Different periods have different names.
Place Value: The value of a digit based on its position in a number.
Place Value Chart: A tool used to identify the position of digits in a number.
Number Forms
Whole Number Notation
Natural Numbers: The counting numbers that range from one to infinity.
Whole Numbers: Zero and the natural numbers.
Standard Notation: Numbers are represented by digits with commas located in every three positions.
Expanded Notation: A way to express a number by separating each digit and its place value name. A plus sign "+" separates each digit and place value name.
Word Name: The spoken name of a number.
​Number Expanded Notation: An alternative form of expanded notation that uses zeros instead of place value names. A plus sign "+" separates each number in number expanded notation.
Integers
Integers
Integers: Whole numbers and their additive inverses.
Additive Inverse: The opposite of a given number. When a number is added to its additive inverse, it equals zero. Since zero is neither positive nor negative, it has no additive inverse and is not a positive or negative integer.
Positive Integers: The natural numbers that are right of zero that range from one to infinity.
Negative Integers: The additive inverses of the natural numbers that are left of zero and that range from negative one to negative infinity.
Fractions
Fractions
Fractions: Numbers composed of a numerator, denominator, and fraction bar.
Numerator: The number above the fraction bar. A numerator is equivalent to a dividend.
Denominator: The number below the fraction bar. A denominator is equivalent to a divisor.
Fraction Bar: The bar that separates the numerator from the denominator. The bar is equivalent to a division sign "÷" and enacts the same operation.
Improper Fractions: Fractions with an equal or greater number in the numerator as compared to the denominator.
Complete Fractions: Improper fractions with the same number in the numerator and denominator. Complete fractions are equal to one.
Visual Representation (Fractions): Fractions are often represented visually, such as portions of a circle, pieces of a block, or divisions of a line.
Equivalent Fractions: Fractions that represent the same value but have different numerators and denominators.
Equivalent fractions can be determined by the cross-products test, long division, or fraction bars.
Fraction Bars: A tool used to visually determine equivalent fractions.
Equivalent fractions can be produced by multiplying the fraction of interest by a fraction with the same number in the numerator and denominator. Because of the multiplicative identity, this multiplication operation is equivalent to multiplying the fraction of interest by one.
Simplified Fraction: A fraction that contains a denominator and numerator that only share one as a common factor.
When answering questions that require a fraction answer, the fraction must be simplified unless instructed not to. To simplify a fraction, you must divide the numerator and denominator by their greatest common factor (GCF).
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​Sequential Simplifying: A multi-step approach to simplifying a fraction.
Alternatively, any shared natural number factor can be used to divide the numerator and denominator; however, the new fraction may need further simplification.
Fractions containing decimals do not typically require simplification unless otherwise requested by a question.
Any integer or decimal value can be converted into a fraction by putting it over one. This conversion becomes essential when completing equations that are partially composed of fractions.
Mixed Numerals
Mixed Numbers
Mixed Numerals: Numbers represented by a whole number and a fraction less than one.​ Mixed numerals are commonly used to describe the length of measurements made with devices such as rulers or tape measures.
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Mixed numerals represent the sum of a whole number added to a fraction less than one.
Mixed numeral notation is an alternative form of improper fraction notation.
Visual Representation (Mixed Numerals): Mixed numerals are always represented by one or multiple complete models and a single incomplete model.
Ratios
Ratios
Ratio: The quotient of two quantities with units. A ratio can be written in fraction or colon notation.
Rate: A ratio that compares two different kinds of units.
Unit: Represents quantities or measurements such as the number of items, grams, meters, points scored, or seconds.
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There are various ways of representing a ratio. Ratios are written either in colon notation or fraction notation. Ratios use "to" to describe the division bar in word form. Rates use "per" to describe the division bar in word form.
Unlike normal fractions, ratios may not always need to be simplified. Depending on the question and units, simplifying a ratio may result in losing statistical and descriptive data. To determine if a ratio requires simplification, check the requirements and context of the question.
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Ratios are key to unit conversions.
Addition
Addition
Addition: The process of combining numbers.
Addend: The number being added. In a visual context, a group of blocks being combined.
Sum: The number that results from combining the addends. In a visual context, the total number of blocks.
Addition equations should never be completed horizontally. Completing an addition equation horizontally disregards place values, which can result in errors when calculating. Instead, addition equations should be changed into a top-to-bottom format.
To test the validity of addition equations, check if the inverse operation (subtraction) is also true.
When adding multidigit numbers, the numbers should be aligned. An easy method for aligning multidigit addends is to align them by decimals. After the addends are aligned, any place value without a digit can be filled in with a zero.
Addition begins with the rightmost digits of the addends and moves leftwards.
Carrying: An addition process used if the sum of two digits is greater than nine. Regarding the sum of two digits, the digit in the one's place remains, and the digit in the ten's place becomes an additional addend for the place value to the left.
To add or subtract fractions, the fractions must share a common denominator. To find the common denominator, the least common multiple (LCM) of the denominators must be deduced.
Multiple: The product of a number and an integer.
Least Common Multiple (LCM): The smallest number that is a multiple of two or more numbers of interest.
Least Common Denominator (LCD): The least common multiple of the denominators of two or more fractions.
Mixed numerals should be converted to their improper fraction form before beginning the addition or subtraction process for fractions.
The commutative and associative laws of addition state that addends can be added in any order, and the sum will remain constant.
Commutative Law of Addition: When two numbers are added, changing the order of the addends does not affect the sum.
Associative Law of Addition: When three or more numbers are added, changing the order of the addends does not affect the sum.
Subtraction
Subtraction: The process of determining the difference between two numbers.
Minuend: The number that is subtracted from. In a visual context, the original total number of blocks.
Subtrahend: The subtracting number. In a visual context, the number of blocks removed.
Difference: The number that results from subtracting one number from another. In a visual context, the number of blocks remaining.
Subtraction equations should never be completed horizontally. Completing a subtraction equation horizontally disregards place values, which can result in errors when calculating. Instead, subtraction equations should be changed into a top-to-bottom format.
To test the validity of subtraction equations, check if the inverse operation (addition) is also true.
When subtracting multidigit numbers, the numbers should be aligned. An easy method for aligning multidigit minuends and subtrahends is to align them by decimals. After the minuend and subtrahend are aligned, any place value without a digit can be filled in with a zero.
Subtraction begins with the rightmost digits of the minuend and subtrahend and moves leftwards.
Division
Division
Division: The process of splitting one number by another. Division can be thought of as repeated subtraction.
Dividend: The number that is split. In a visual context, the total number of blocks.
Divisor: The number dividing another number. In a visual context, the amount of blocks placed into each group.
Quotient: The number that results when one number is divided by another. In a visual context, the total number of groups formed.
To test the validity of division equations, check if the inverse operation (multiplication) is also true.
Additionally, knowledge of multiplication facts can be an alternative method for determining the quotient. This can be done by rearranging the division equation into its inverse operation, multiplication.
When numbers can't be evenly divided, the quotient will contain a remainder, a fraction, or a decimal.
Remainder: The leftover value in a division problem.
Division problems containing a remainder can be checked using the inverse operation (multiplication) and addition to account for the remainder.
Instead of having a remainder, the quotient may contain a fraction or decimal.
Long division is a procedure that is especially useful for solving equations containing large numbers or decimals; however, long division can be used for any division equation.
Division may result in a non-terminating decimal or a decimal that reaches distant place values. Rounding these quotients to the thousandth place when possible gives a reasonable estimate of the actual value.
​To divide by a fraction, multiply by the reciprocal.
Reciprocal (Fractions): The reciprocal of a fraction can be found by swapping the numerator and denominator. If the reciprocal is multiplied by the original fraction, the product is always one.
Similarly, to divide by a mixed numeral, convert the mixed numeral to its improper fraction form and multiply by its reciprocal.
Complex Fraction: A fraction in which the numerator and/or the denominator contain one or more fractions. Complex fractions are solved by multiplying by the divisor's reciprocal.
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When zero is a divisor in a division equation, it will always result in an undefined quotient. When dividing by zero, the logical consequence would be a quotient of zero; however, the inverse operation would result in any desired number. Thus, zero is excluded as a divisor and always results in an undefined quotient.
Factorization
Factorization
Factors: Natural number multipliers that result in a natural number product.
(natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12...)
If a number is a factor of a product, it also means that the product is a multiple of the factor.
Factorization: A natural number expressed as the product of two or more natural numbers.
Only natural numbers are considered when determining a number's factors; multipliers containing decimals, fractions, negative numbers, or remainders are not considered factors of a number.
Being able to determine all the factors of a number is essential. This skill makes multiplying, dividing, and simplifying fractions easier.
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To identify factors, one must determine natural number multipliers that equal the product or divide the product by natural numbers to check for factors. Either approach, division or multiplication, is valid.
The product itself and one will always be factors. This is the case since one is the multiplicative identity and always equals the number it is multiplied by.
Prime Numbers: Numbers that only have two factors: one and itself. Two is the only even prime number.
Composite Numbers: Numbers that have more than two factors.
Multiplication Chart: A mathematics tool that shows the multiples of a set of numbers. This table assists in determining the factors of a number.
Divisibility Tests: Tests that circumvent long division and allow one to check if an integer is divisible by 2, 3, 5, 6, 9, or 10. These tests assists in determining the factors of a number.
Prime Factorization: The sequential factorization of a composite number and its factors that results in a unique set of prime numbers. The prime numbers produced by prime factorization multiply to produce the original number factorized.
Factor Tree: A visual prime factorization method that involves factoring composite numbers until only prime numbers remain.
​​​Prime factorization can be used as a method to find the least common multiple for a set of numbers.
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